A discretization of Volterra integrals equations of the third kind with weakly singular kernels

In this paper we propose a method of piecewise constant approximation for the solution of ill-posed third kind Volterra equations $$p(t)z(t) + int_0^t frac{h(t,tau)}{(t-tau)^{1-alpha}}z(tau) dtau = f(t),quad tin[0,1],enspace 0<alpha< 1.$$ Here $p(t)$ vanishes on some subset of $[t_1,t_2]subset[0,1]$ and mbox{$p(t) <delta$} for {mbox{$tin[t_1,t_2]$}}, where $delta$ is a sufficiently small positive number. The proposed method gives the accuracy $O(delta^{2nu/(2nu+1)})$ with respect to the mbox{$L_2$-norm,} where $nu$ is the parameter of sourcewise representation of the exact solution on $[t_1,t_2]$, and uses no more than $O(delta^{-(2-lambda)/alpha} log^{2+1/alpha}frac{1}{delta})$ values of Galer-kin functionals, where $lambdain(0,1/2)$ is determined in the act of choosing the regularization parameter within the framework of Morozov's discrepancy principle

A discretization of Volterra integrals equations of the third kind with weakly singular kernels

In this paper we propose a method of piecewise constant approximation for the solution of ill-posed third kind Volterra equations $$p(t)z(t) + int_0^t frac{h(t,tau)}{(t-tau)^{1-alpha}}z(tau) dtau = f(t),quad tin[0,1],enspace 0<alpha< 1.$$ Here $p(t)$ vanishes on some subset of $[t_1,t_2]subset[0,1]$ and mbox{$p(t) <delta$} for {mbox{$tin[t_1,t_2]$}}, where $delta$ is a sufficiently small positive number. The proposed method gives the accuracy $O(delta^{2nu/(2nu+1)})$ with respect to the mbox{$L_2$-norm,} where $nu$ is the parameter of sourcewise representation of the exact solution on $[t_1,t_2]$, and uses no more than $O(delta^{-(2-lambda)/alpha} log^{2+1/alpha}frac{1}{delta})$ values of Galer-kin functionals, where $lambdain(0,1/2)$ is determined in the act of choosing the regularization parameter within the framework of Morozov's discrepancy principle

A discretization of Volterra integrals equations of the third kind with weakly singular kernels

In this paper we propose a method of piecewise constant approximation for the solution of ill-posed third kind Volterra equations $$p(t)z(t) + int_0^t frac{h(t,tau)}{(t-tau)^{1-alpha}}z(tau) dtau = f(t),quad tin[0,1],enspace 0<alpha< 1.$$ Here $p(t)$ vanishes on some subset of $[t_1,t_2]subset[0,1]$ and mbox{$p(t) <delta$} for {mbox{$tin[t_1,t_2]$}}, where $delta$ is a sufficiently small positive number. The proposed method gives the accuracy $O(delta^{2nu/(2nu+1)})$ with respect to the mbox{$L_2$-norm,} where $nu$ is the parameter of sourcewise representation of the exact solution on $[t_1,t_2]$, and uses no more than $O(delta^{-(2-lambda)/alpha} log^{2+1/alpha}frac{1}{delta})$ values of Galer-kin functionals, where $lambdain(0,1/2)$ is determined in the act of choosing the regularization parameter within the framework of Morozov's discrepancy principle

Reports about 8 selected benchmark cases of model hierarchies : Deliverable number: D5.1 - Version 0.1

Based on the multitude of industrial applications, benchmarks for model hierarchies will be created that will form a basis for the interdisciplinary research and for the training programme. These will be equipped with publically available data and will be used for training in modelling, model testing, reduced order modelling, error estimation, efficiency optimization in algorithmic approaches, and testing of the generated MSO/MOR software. The present document includes the description about the selection of (at least) eight benchmark cases of model hierarchies.EC/H2020/765374/EU/Reduced Order Modelling, Simulation and Optimization of Coupled Systems/ROMSO

Modelling, Simulation and Data Analysis in Acoustical Problems

Modelling and simulation in acoustics is currently gaining importance. In fact, with the development and improvement of innovative computational techniques and with the growing need for predictive models, an impressive boost has been observed in several research and application areas, such as noise control, indoor acoustics, and industrial applications. This led us to the proposal of a special issue about “Modelling, Simulation and Data Analysis in Acoustical Problems”, as we believe in the importance of these topics in modern acoustics’ studies. In total, 81 papers were submitted and 33 of them were published, with an acceptance rate of 37.5%. According to the number of papers submitted, it can be affirmed that this is a trending topic in the scientific and academic community and this special issue will try to provide a future reference for the research that will be developed in coming years